By: Cole Srebro, Dillan Patel, Jerry Langan

Deductive Structure -a system, of thought in which conclusions are justified by means of previously proved or assumed statements. 4 Elements of Deductive Structure Undefined Terms (Points, Rays, Lines, Segments) Assumptions known as postulates Definitions (Reversible) Theorems and other conclusions (Can be proved)

Deductive Structure  Postulate -an unproved assumption › Ex. Through a point not on a line there is exactly one parallel to the given line.  Definition -states the meaning of a term or idea › Ex. If a point is the midpoint of a segment, then it divides the segment into two congruent segments › If a point divides a segment into two congruent segments, then it is the midpoint of the segment. (Reverse)

 Conditional Statement- “if p, then q” p and q are declarative statements. The “if” part of the sentence is called the hypothesis. The “then” part of the sentence is called the conclusion. If “p” the “q” can also be written p=>q (p implies q).  Converse- the converse of p=>q is q=>p. To write the converse of a conditional statement you reverse parts p and q.  Biconditional-if a conditional statement and its converse are both true, the statement is said to be biconditional.

 Write the converse of each of the following statements:  If C, then DIf D, then C  If peppers, then spicyIf spicy, then peppers  All cloudy days are depressing. Therefore, since I was depressed on Friday, Friday was cloudy. It could have been cloudy but he could have been depressed from something else.

1. Jim is a barber. Everybody who gets his hair cut by Jim gets a good haircut. Austin got a good haircut. What can you deduce about Austin? 2. All dogs are mammals, and all mammals are vertebrates. Shaggy is a dog. What can be deduced about shaggy? 3. When the sun shines, the grass grows. When the grass grows, it needs to be cut. The sun shines. What can you deduce about the grass?

Answers to Practice Problems(1.7) 1. Nothing. Just because Austin got a good haircut does not mean that Jim cut his hair. This is always possible, but nothing can be deduced from the situation. 2. Shaggy is a mammal and a vertebrate. 3. It needs to be cut.

Statements of Logic Negation-the negation of any statement “p” is the statement “not p”. The symbol for “not p” is ~p. In general, ~ ~p=p, [not (not p)=p]. Converse- (if q, then p). Review from section 1.7 Inverse- (if ~p, then ~ q). Contrapositive- (if ~q, then ~p).

Sample Problems Ex. Conditional: If you live in Los Angeles, then you live in California. (True) Converse: If you live in California, then you live in Los Angeles. (False) Inverse: If you don’t live in Los Angeles, then you don’t live in California. (False) Contrapositive: If you don’t live in California, then you don’t live in Los Angeles. (True)

Geometry Sample Problem  Conditional: If a ray divides and angle into two congruent angles, then it bisects the angle. (True)  Converse: If a ray bisects an angle, then it divides the angle into two congruent angles. (True)  Inverse: If a ray doesn’t divide and angle into two congruent angles, then it doesn’t bisect the angle. (True)  Contrapositive: If a ray doesn’t bisect an angle, then it doesn’t divide the angle into two congruent angles. (True)

Statements of Logic  The sample problems on the previous slides display this important theorem.  Theorem 3: If a conditional statement is true, then the contrapositive of the statement is also true. (If p, then q if ~q, then ~p.)

 Chains of reasoning  Ex. If p => q and q=> r, then p => r. This is called the chain rule, and a series of conditional statements so connected is known as a chain of reasoning.  Procedure to form a chain of reasoning and write a concluding statement: 1. Make a list of contrapositives to the given conditional statements. 2. Start the chain with a variable used only once in the given conditionals. 3. Continue making the chain using both the list of conditionals and the list of contrapositives until the chain is complete. 4. Write a concluding statement.

 Ex.  Conditionals Contrapositives g=>e ~e=>~g ~t=>w ~w=>t t=>~e e=>~t  Chain: g=>e=>~t=>w or ~w=>t=>~e=>~g  Concluding statement: g=>w or ~w=>~g

Practice Problems(1.8) 1. Chain of Reasoning ˜x=>˜w u=>t x=>˜t v=>w 2. Write the converse, the inverse, and the contrapositive of each statement: If a point is the midpoint of a segment, then it divides the segment into two congruent segments.

1. Chain of Reasoning ˜x=>˜w w=>x u=>t ˜t=>˜u x=>˜t t=>˜x v=>w ˜w=>˜v Chain…u=>t=>˜x=>˜w=>˜v Conclusion … u=>˜v

Answers to Practice Problems Continued (1.8) Answers to Practice Problems Continued (1.8) 2. Converse: If a point divides a segment into two congruent segments, then it is the midpoint of that segment. Inverse: If a point is not the midpoint of a segment, then it doesn’t divide that segment into two congruent segments. Contrapositive: If a point doesn’t divide a segment into two congruent segments then it is not the midpoint of that segment.

W ORKS C ITED Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Littell & Company, Print. Sparknotes. Sparknotes. Web. 17 January 2010.